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1
Basic Concepts of Robust Design
Optimization (RDO)
WOST15 2018
Veit Bayer
Christian Bucher
Dynardo Weimar Vienna
2Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview
Introduction
Definitions of Probability
Random Variables and Vectors
Simulation
Estimation
Robustness and Reliability Analysis
Variance-based sigma levels reliability-based
Methods of Reliability Analysis
Examples
3Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Introduction
4Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
Design ImprovementOptimize design performance
copy Dynardo GmbH
5Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
1 Search optima with
flat surrounding
(stability)
6Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
2 Keep safety distance
from infeasible domain
(safety quality)
7Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Probability
8Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Consider a sample space the set of all possible events (or all possible outcomes of basic variables)
bull Event that one realization of parameters falls into subdomain
Kolmogorov axioms
Events and Probabilities
9Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Complimentary events
bull An event can happen ( ) or not happen ( )
bull Complimentary events cannot happen at the same time
Events and Probabilities
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
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Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
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bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
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Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
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Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
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Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
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Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
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Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
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Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
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Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
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Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
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Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
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Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
2Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview
Introduction
Definitions of Probability
Random Variables and Vectors
Simulation
Estimation
Robustness and Reliability Analysis
Variance-based sigma levels reliability-based
Methods of Reliability Analysis
Examples
3Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Introduction
4Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
Design ImprovementOptimize design performance
copy Dynardo GmbH
5Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
1 Search optima with
flat surrounding
(stability)
6Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
2 Keep safety distance
from infeasible domain
(safety quality)
7Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Probability
8Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Consider a sample space the set of all possible events (or all possible outcomes of basic variables)
bull Event that one realization of parameters falls into subdomain
Kolmogorov axioms
Events and Probabilities
9Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Complimentary events
bull An event can happen ( ) or not happen ( )
bull Complimentary events cannot happen at the same time
Events and Probabilities
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
3Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Introduction
4Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
Design ImprovementOptimize design performance
copy Dynardo GmbH
5Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
1 Search optima with
flat surrounding
(stability)
6Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
2 Keep safety distance
from infeasible domain
(safety quality)
7Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Probability
8Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Consider a sample space the set of all possible events (or all possible outcomes of basic variables)
bull Event that one realization of parameters falls into subdomain
Kolmogorov axioms
Events and Probabilities
9Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Complimentary events
bull An event can happen ( ) or not happen ( )
bull Complimentary events cannot happen at the same time
Events and Probabilities
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
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Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
4Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
Design ImprovementOptimize design performance
copy Dynardo GmbH
5Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
1 Search optima with
flat surrounding
(stability)
6Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
2 Keep safety distance
from infeasible domain
(safety quality)
7Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Probability
8Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Consider a sample space the set of all possible events (or all possible outcomes of basic variables)
bull Event that one realization of parameters falls into subdomain
Kolmogorov axioms
Events and Probabilities
9Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Complimentary events
bull An event can happen ( ) or not happen ( )
bull Complimentary events cannot happen at the same time
Events and Probabilities
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
5Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
1 Search optima with
flat surrounding
(stability)
6Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
2 Keep safety distance
from infeasible domain
(safety quality)
7Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Probability
8Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Consider a sample space the set of all possible events (or all possible outcomes of basic variables)
bull Event that one realization of parameters falls into subdomain
Kolmogorov axioms
Events and Probabilities
9Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Complimentary events
bull An event can happen ( ) or not happen ( )
bull Complimentary events cannot happen at the same time
Events and Probabilities
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
6Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robust Design Optimization (RDO) optimizes the design performance while taking into account scatter of design (optimization) variables and other tolerances or uncertainties
bull As a consequence of input scatter the location of the optima as well as the contour lines of constraints may vary
Uncertainties in Optimization
2 Keep safety distance
from infeasible domain
(safety quality)
7Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Probability
8Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Consider a sample space the set of all possible events (or all possible outcomes of basic variables)
bull Event that one realization of parameters falls into subdomain
Kolmogorov axioms
Events and Probabilities
9Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Complimentary events
bull An event can happen ( ) or not happen ( )
bull Complimentary events cannot happen at the same time
Events and Probabilities
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
7Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Probability
8Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Consider a sample space the set of all possible events (or all possible outcomes of basic variables)
bull Event that one realization of parameters falls into subdomain
Kolmogorov axioms
Events and Probabilities
9Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Complimentary events
bull An event can happen ( ) or not happen ( )
bull Complimentary events cannot happen at the same time
Events and Probabilities
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
8Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Consider a sample space the set of all possible events (or all possible outcomes of basic variables)
bull Event that one realization of parameters falls into subdomain
Kolmogorov axioms
Events and Probabilities
9Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Complimentary events
bull An event can happen ( ) or not happen ( )
bull Complimentary events cannot happen at the same time
Events and Probabilities
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
9Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Complimentary events
bull An event can happen ( ) or not happen ( )
bull Complimentary events cannot happen at the same time
Events and Probabilities
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
10Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Conditional Probability
Independence
Events and Probabilities
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
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Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
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Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
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Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
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Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
11Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Total Probability
Bayesrsquo Theorem
Purpose Testing model updating conclude from a measurement eg to product safety or quality
Decomposition of Event Space
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
12Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Random Variables
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
13Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Statistical Characterization of Random Variables
bull Expectation operator
bull Mean value
bull Variance
bull Coefficient of variation
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
14Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Probability Distribution and Density Function
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
15Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Distribution Types
Uniform Normal Log-normal
Exponential Weibull Rayleigh
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
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Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
16Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Statistical Characterization of Random Vectors
bull Arbitrary number k of random variables can be arranged in a vector
bull Mean value vector
bull Coefficient of correlation between two random variables
bull Covariance matrix of a random vector
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
17Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Random generators produce numbers uniformly distributed in [01]
bull Mapping to prescribed marginal distribution
copy Dynardo GmbH
Simulation of Random Variables
fU
u FX
fX
x
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
18Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull For each random variable the original marginal distribution is
transformed to an uncorrelated standard normal variable by the CDF
bull Assume a correlated joint Normal distribution for the random vector
bull Iterate the correlation coefficients of Zi Zj to match the original ones
copy Dynardo GmbH
Simulation of Random Vectors
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
19Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Example
copy Dynardo GmbH
Simulation of Random Vectors
Standard normal space Original space
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
20Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Estimation
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
21Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Estimate an unknown parameter from independent observations
bull Example mean value
bull Consistency
bull (Asymptotic) Unbiasedness
Remarks
bull The true parameter is usu not known the available information is the
sample
bull Any estimate from a finite sample contains statistical uncertainty
which can be reduced by an increased sample size
copy Dynardo GmbH
Estimation
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
22Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull An estimator from a random sample is a random variable by itself
bull Variance of the estimator
bull Estimator for variance
bull Estimate the variance of the estimator
serves to assess the confidence of the estimate
copy Dynardo GmbH
Estimator Variance
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
23Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Statistical error (or standard error) of the estimator
bull If the distribution of the error is known (eg assume Normal)
then the confidence interval can be established
copy Dynardo GmbH
Confidence Interval
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
24Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robustness Analysis
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
25Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Intuitively The performance of a robust design is largely unaffected by random perturbations
bull Variance indicator The coefficient of variation (CV) of the objective function andor constraint values is not greater than the CV of the input variables
bull Sigma level The interval mean+- sigma level does not reach an undesired performance (eg design for six-sigma)
bull Probability indicator The probability of reaching undesired performance is smaller than an acceptable value
How to Define the Robustness of a Design
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
26Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Robustness in terms of limits
bull Safety margin (sigma level) of one or more responses y
bull Reliability (failure probability) with respect to given limit state
Robustness in terms of stability
bull Performance (objective) of robust optimum is less sensitive to input uncertainties
bull Minimization of statistical evaluation of objective function f (eg minimize mean andor standard deviation)
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
27Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Taguchi loss functions
bull Target value m is optimal (k scaling factor for costs)
bull Minimum is optimal (requires positive objective)
bull Maximum is optimal (requires strictly positive objective)
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
28Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Variance based Robustness Analysis
1) Define the robustness space using scatter range distribution and correlation
2) Scan the robustness space by producing and evaluating ndesigns
3) Check the variation 4) Check the
explainability of the model
5) Identify the most important scattering variables
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
29Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Exceedance Probability
bull Probability of reaching values above a limit for Gaussian distribution
m x
fX(x)
x
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
30Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Sigma Level vs Failure Probability
bull The sigma level can be used to estimate the probability of exceeding
a certain response limit
bull Since the distribution type of the response is generally unknown
this estimate may be very inaccurate for small probabilities
(sigma levels larger than 3)
bull The sigma level deals with single limit values whereas the failure
probability quantifies the event that any of several limits is exceeded
Reliability analysis should be applied to proof the required safety level
Distribution Required sigma level (CV=20)
pF = 10-2 pF = 10-3 pF = 10-6
Normal 232 309 475
Log-normal 277 404 757
Rayleigh 272 376 611
Weibull 203 254 349
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
31Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Example Optimized Damped Oscillator
bull Robustness evaluation at
the deterministic optimum
bull Mass m damping ratio D stiffness k and initial kinetic energy Ekin
taken as normally distributed random variables
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
32Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull Robustness analysis with respect to damped eigen-frequency and
maximum amplitude
ndash Check CV of objective and constraints
ndash Check if safety constraint safety = 85 rads
is outside of 45 level
ndash Check importance of input variables
ndash Check explainability by MOPCoP
Example Damped OscillatorVariance based Robustness Analysis
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
33Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Constraint equation (omega)
bull CoD and CoP is 100
bull k is most important m is minor
bull Mean is close to deterministic
value
bull CV is 27
bull Safety limit is 238 which is
smaller as the required 45
Optimum is not robust in terms of
the constraint condition
Example Damped Oscillator
238
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
34Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Objective function (xmax)
bull CoD and CoP is 92
bull D is most important Ekin
and k are minor important
bull Mean is not close to
deterministic value
bull CV is 110
Optimum is not robust in terms
of the objective function
Example Damped Oscillator
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
35Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Reliability Analysis
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
36Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Concept of Safety
bull Failure occurs if loading S exceeds the resistance R
bull Probability of failure
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
37Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbHcopy Dynardo GmbH
Partial Safety Factors
bull Definition of characteristical values for loading Sk and resistance Rk
bull Design values are obtained by
using partial safety factors
bull Final safety proof
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
38Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
bull The complementary of reliability the probability of failure PF is computed (for numerical reasons)
bull PF is the probability of the event that a set of (stochastic) input parameters leads to an inadmissible state of the analyzed system
(eg exceedance of allowable stress)
bull The limit state function g(x) separates the random variable space Xinto a safe domain g(x) gt 0 and failure domain g(x) le 0
bull Multiple failure criteria (limit state functions) are possible
bull Series system
fails if one single component fails
g(x) = mini (gi (x))
bull Parallel system
fails if all components fail
g(x) = maxi (gi (x))
copy Dynardo GmbH
Reliability Analysis
FF
G
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
39Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Failure Probability
bull The probability of failure is the integral of the joint probability density
function over the failure domain
bull By introducing an indicator function
I(g(x)) = 1 if (g(x)) lt 0 I(g(x)) = 0 else
this can be computed as the expected value of I
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
40Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Monte Carlo Simulation
bull Robust for arbitrary limit state functions
bull Confidence of the estimate is very low for small failure probabilities
Sigma level le 2
Independent of number of random variables
X1
X2
g=0
Sigma
level
PF N for cov(PF) = 10
2 23E-2 4 400
3 13E-3 74 000
45 34E-6 29 500 000
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
41Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
First Order Reliability Method (FORM)
bull Operates in the space of
standardized Gaussian variables
bull Search for failure point with
maximum probability density
(design point)
bull Equals the point in U on the limit state surface with minimal
distance to origin
bull Limit state function is linearized
around design point
bull Then failure probability can be
calculated analytically
bull Distance to origin (in U) is called
reliability index b
bull Can be interpreted as
generalization of sigma level
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
42Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling
bull Guide the sampling by making use of information about the failure
domain in order to increase the amount of failure events
bull To warrant correct statistics each sample is weighted by the ratio of
original to sampling density
bull Different strategies exist to estimate an ldquooptimalrdquo sampling density
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
43Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Importance Sampling Using Desing Point (ISPUD)
bull Based on FORM
bull Sampling density is centered at the design point
Requires continuously differentiable limit state function
Multiple design points (local minima) are not supported
May be able to mitigate error due to linearization in FORM
(oscillating limit state surface)
Moderate number of random variables
g(X) = 0
design point
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
44Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Importance Sampling
bull Sampling density is defined by mean value vector and covariance
matrix of samples in the failure domain
bull Search for dominant failure region by 2-3 sampling iterations
Applicable for non-smooth and even discontinuous limit state functions
Limited to small to medium number of random variables
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
45Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Directional Sampling
bull Radial search for multiple ldquostar-shapedrdquo failure regions
Applicable for non-smooth and even discontinuous limit state functions
Limited to small number of random variables
Few unsuccessful solver calls possible (as long as search is successful)
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
46Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Adaptive Response Surface Method
bull The limit state function is approximated by an Adaptive Response
Surface Method using a Moving Least Squares model
bull Directional Sampling is performed on the Response Surface
bull Additional supports are added near the limit state surface in regions of
high probability density
Applicable to a wide range of limit state functions
Efficient for a moderately high number of random variables
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
47Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Overview of Methods
Recommended area of application
Approach Non-linearity Failure domains No parameters No solver runs
Monte Carlo
Simulation
arbitrary arbitrary many gt10^4 (3 sigma)
gt10^7 (5 sigma)
Directional
Sampling
arbitrary arbitrary lt= 10 1000-5000
Adaptive Importance
Sampling
arbitrary one dominant lt= 10 500-1000
FORM SORM
ISPUD
monotonic one dominant lt= 20 200-500
Adaptive Response
Surface Method
continuous few dominant lt= 20 200-500
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
48Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVerification of Robust Design by Reliability Analysis
bull Safety margin of 45 is equivalent to a failure probability of 3410-6
if responses were normally distributed
Reliability
Method Samples Failure probability Error Beta
FORM 65 1310-6 - 47
Adaptive Sampling 1500 1310-6 8410-8 47
Directional Sampling 600 1310-6 4910-7 47
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
49Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for Best Practice
bull If there is no qualified information about the random parametersonly variance-based robustness evaluation is justified
bull Results with high sigma levels must be verified by reliability analysis
bull Choose proper reliability method due to dimension reliability level solver behavior
bull Reliability results shall be confirmed by a second method
bull When a reduced parameter set is used a confirmation with full parameter set is required
bull Use MOP (based on robustness samples) in order to
bull Monitor sampling
bull Monitor solver behavior
bull Analyze cause for non-robustness
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
50Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Recommendations for best practice
bull The Robustness Wizard of optiSLang guides through the setup and helps with the decision for the method based on
bull Knowledge about uncertainty
bull Number of failed designs
bull Solver behavior
bull Sigma level
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
51Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Robust Design Optimization
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
52Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Model CalibrationsIdentify important model parameter
for the best fit between simulation
and measurement
Design ImprovementOptimize design performance
Design QualityEnsure design robustness
and reliability
Design QualityEnsure design robustness
and reliability
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
Design UnderstandingInvestigate parameter sensitivities
reduce complexity and
generate best possible meta models
CAE-Data
Measurement
Data
Robust Design
copy Dynardo GmbH
Design ImprovementOptimize design performance
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
53Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Iterative Robust Design Optimization
bull Decoupled optimization and
robustnessreliability analysis
bull For each optimization run the
safety margins are adjusted for
the critical model responses
bull Applicable to variance- and
reliability-based RDO
In our implementation variance-
based robustness analysis is
used inside the iteration and a
final reliability proof is performed
for the final design
Definition of
design and
stochastic
variables
Sensitivity
analysis
Design
failure
Update
constraints
Deterministic
optimization
Variance-
based
robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
54Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Iterative RDO
bull Safety margin of 45s to safety limit safety=85 rads
Sigma level requires (safety-mean) ge 45
Deterministic constraint is modified iteratively
Step 1 Step 2 Step 3 hellip
Optimization (global ARSM) Robustness (100 ALHS)
Constraint m k xmax Mean Sigma Sigma level
le 8 078 500 799 025 799 022 233
le 7 104 500 694 029 694 019 82
le 763 086 491 756 028 755 020 466
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
55Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled Robust Design Optimization
bull Fully coupled optimization and robustnessreliability analysis
bull For each design during the optimization procedure (nominal design)
the robustnessreliability analysis is performed
bull Applicable to variance- reliability- and Taguchi-based RDO
Our efficient implementation uses small sample variance-based
robustness measures during the optimization and a final
(more accurate) reliability proof
But still the procedure is often not applicable to complex CAE models
Definition of
design and
stochastic
variables
Sensitivity
analysisOptimization
Robustness
evaluation
Final
reliability
proof
Optimal and
robust
design
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
56Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Coupled RDO in optiSLang
bull Nested loop enables the coupled RDO
bull Optimizer has to handle statistical
errors of inner robustness analysis
bull Sigma level as constraint
See tutorial HelpTutorialsOscillatorOscillator_Robustness
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
57Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
Example Damped OscillatorVariance-based Fully Coupled RDO
bull ARSM + robustness analysis
bull For each design 1+20 solver runs
bull Definition of robustness constraint for optimization procedure (safety-mean)-45s ge 0
Robust Design Optimization (ARSM+LHS)
m k xmax Mean Sigma Sigma level
ARSM+20LHS 087 500 028 76 020 45
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
58Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
copy Dynardo GmbH
bull Approximation of model responses in
mixed optimizationstochastic space
bull Simultaneous RDO is performed on
a global response surface
bull Applicable to variance- reliability-
and Taguchi-based RDO
bull Approximation quality significantly
influences RDO results
Final robustnessreliability proof
is required
bull Pure stochastic variables have small
influence compared to design variables
Important local effects in the stochastic
space may be not represented
RDO on Global Response Surface
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
59Basics Concepts of Robust Design Optimization
WOST 15 Weimar June 21 2018
C Bucher Computational Analysis of Randomness in Structural
Mechanics Taylor amp Francis 2009
copy Dynardo GmbH
Further Reading
top related